Continuum percolation with steps in an annulus

2004/11/01 by Paul Balister, Béla Bollobás, Bela Bollobas +1 · 2 citations
Mathematics · #Limits and Structures in Graph Theory #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60K35 #msc:82B43.

paper · pdf · doi:10.1214/105051604000000891

published as Annals of Applied Probability 2004, Vol. 14, No. 4, 1869-1879 · Published at http://dx.doi.org/10.1214/105051604000000891 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2004/11/01 · arxiv created 2005/03/24 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let A be the annulus in ℝ2 centered at the origin with inner and outer radii r(1−ɛ) and r, respectively. Place points xi in ℝ2 according to a Poisson process with intensity 1 and let \mathcal G_A be the random graph with vertex set xi and edges xixj whenever xi−xj∈A. We show that if the area of A is large, then \mathcal G_A almost surely has an infinite component. Moreover, if we fix ɛ, increase r and let nc=nc(ɛ) be the area of A when this infinite component appears, then nc→1 as ɛ→0. This is in contrast to the case of a “square” annulus where we show that nc is bounded away from 1.

Citations

• Random Graphs
• Nearest neighbor and hard sphere models in continuum percolation

Cited by

• Line-of-Sight Percolation
• Spread‐out percolation in ℝd

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